Many confuse the term 'old covenant' with the label 'Old Testament', thinking all the Old Testament Books were about the 'old covenant'. Not so. The old covenant involved laws in ordinances dealing with religious ceremonial worship, rituals, washings, tithes, etc. That was not part of God's Ten Commandments, nor many other laws of God's creation.
What Lord Jesus nailed to His cross specifically was the "handwriting of ordinances" (Colossians 2:14). That does not mean all of God's laws. And the Christian can easily see this with what Apostle Paul taught in 1 Timothy 1 about God's laws...
1 Tim 1:8-11
8 But we know that the law is good, if a man use it lawfully;
9 Knowing this, that the law is not made for a righteous man, but for the lawless and disobedient, for the ungodly and for sinners, for unholy and profane, for murderers of fathers and murderers of mothers, for manslayers,
10 For whoremongers, for them that defile themselves with mankind, for menstealers, for liars, for perjured persons, and if there be any other thing that is contrary to sound doctrine;
11 According to the glorious gospel of the blessed God, which was committed to my trust.
KJV
Paul gives warning for breaking certain of God's laws also in 1 Corinthians 6:9-10, and also in Galatians 5. Paul even says there, those who follow such sins will not enter into the Kingdom of God.
So many of God's laws are still established under Christ Jesus. And it's important to understand what God's laws were established for, i.e., for the wicked and unholy. That means following the commandments of Jesus and His Apostles means we won't be breaking any of God's laws.
That is actually Paul's Message in the Galatians 5 Chapter about walking by The Spirit, and not by our flesh. If we walk by The Holy Spirit, listening to Him, then 'we' become dead to the law. A danger to be careful of is with some churches that preach this backwards, and instead say that all of the law is now dead under Christ, which is not true per New Covenant doctrine given by Apostle Paul. So careful 'how' you hear.